Final answer:
The potential energy of repulsion per unit area Vr in the context of electric charges and fields is derived from the potential of a point charge V = kQ / r and integrated to obtain the potential energy U = kQq / r or related expressions in specific cases, such as van der Waals forces.
Step-by-step explanation:
The integration equation for potential energy of repulsion per unit area Vr involves the electric potential V of a point charge, which is given by the expression V = kQ / r, where k is Coulomb's constant, Q is the charge, and r is the distance from the charge. If we are considering the potential energy between two charges, we would use the formula U = kQq / r, where q is the second charge.
In the context of a uniform electric field E produced by a potential difference AV, the relationship between electric potential and electric field is given by the equation AV = VAB = E·d, where d is the distance over which the electric field is applied.
For special cases like the van der Waals potential or a wire-cylinder system, the potential energy equations might differ and involve more complex integrals. For instance, the van der Waals force can be found through a Taylor series expansion and then comparing with Hooke's law to approximate the force for small displacements near the equilibrium position r = Ro.
Integration is important in these contexts to derive expressions for forces, electric fields, and other related physical quantities. An integral solver can be used to perform these complex integrations, which are often encountered in advanced physics problems.